Quantum Simulations promise new insight into complex quantum behaviour, even before universal quantum computers might be able to run arbitrary algorithms (like Shor's or Grover's). Building blocks for quantum computation, summarized within the DeVinzenzo criteria, are provided at high fidelity for single and two qubit gate operations (geometrical Phase gates -- Möelmer Sörensen Cirac Zoller Milburn...) and allowed for proof of principle experiments on quantum teleportation, quantum dense coding, quantum Fourier analysis and quantum error correction using superposition and entangled states. Other fields outside quantum information processing, like metrology (atomic clocks), quantum cryptography might also benefit from the technological developments in Paul trap (or Penning trap) technology (like surface traps spanned by arrays of micro traps) - allowing protocol similar to simulations within optical lattices for atoms, Coulomb blockade... Simulating solid state systems (potentially described by quantum spin Hamiltonians) would allow deeper insight into quantum Hall effect, Spin Peierls, high temperature superconductivity and its relation to triangular lattices and spin frustration and the Bose-Hubbard model, spin-Boson interaction would be most interesting. Investigating Hawking or Unruh radiation within the same simulator used for simulations on the Dirac equation or quantum walks and Schroedinger cat states. Topological defects after quantum phase transitions (in a first step within the quantum Ising model) could be investigated (Zurek). Decoherence might affect the fidelity of the operations but might also be a gadget (Seth Llyod). Adiabaticity of the evolution and the amount of particles needed to be close to the effects occurring at the thermo dynamic limit are related but to be investigated with charged ions, (sympathetically) laser cooled and frozen into Coulomb crystalline structures, e.g. linear chains. Stealing many ideas from Nuclear magnetic resonance experiments, like composite pulses, spin echo within Ramsey spectroscopy to enhance the contrast of Rabi-flopping (oscillations on the Bloch sphere).
Quantum walks are predicted to be of fundamental interest for many applications. Searching for the optimal path might get dramatically boosted in efficiency if one does not have to try out randomly each individual one but all of them simultaneously. We present the experimental realisation of
The quantum Walk of a trap Ion in phase space